Proceedings of the Tenth Asian Congress of Fluid Mechanics
17–21, May 2004, Peradeniya, Srilanka.
FLUX SCALING AND PLUME STRUCTURE
IN HIGH Ra - HIGH Sc TURBULENT
CONVECTION
Baburaj A. Puthenveettil and Jaywant H Arakeri
Department of Mechanical Engineering, Indian Institute of Science,
Bangalore, India 560012
ABSTRACT: The arrangement of brine above water across a micro porous permeable membrane is
used to study high Rayleigh Number(1011 − 1010 ) high Schmidt number(650)turbulent convection. The
flux shows 4/3rd scaling with line plume as the near wall coherent structures. Shifting of multiple large
scale flow cells result in changing near membrane mean shear directions for large aspect ratios. Lower
aspect ratios show single large scale flow cell and constant sense of mean shear.
1.
INTRODUCTION
In Rayleigh - Benard convection , where buoyancy is the only source of motion, the non
dimensional heat flux, Nusselts number (Nu) can be expressed as a function of the independent
non dimensional parameters viz. Rayleigh number (Ra) which expresses the ratio of buoyancy
forces to the dissipative effects, Prandtl number (Pr), a fluid property and Aspect Ratio (AR =
Length/Height), a geometric parameter. When Rayleigh number is high, turbulent convection
occurs with a well-mixed core and surface boundary layers. The classical 4/3rd law which states
that flux ∼ ∆T 4/3 (ie.N u ∼ Ra1/3 ) is obtained from the assumption that transport process is
determined only by the near wall boundary layers and hence is independent of layer height h.
At the same time, as very high Ra implies negligible dissipative effects, dimensional consistency
demands N u ∼ (RaP r)1/2 [2] . The experimentally observed scaling law is N u = K Ran where
n is slightly less than 1/3, resulting in a weak dependence of flux on layer height. At Ra > 10 8
the system is seen to generate a large-scale mean flow which modifies the near wall diffusive
boundary layers by shear effects, and is expected to be the reason for the observed value of n .
Plumes play the major role in transporting heat in this regime from the boundary layer. [13] Various
phenomenological explanations, each involving major assumptions, viz. a plume dominated mixing
zone,[2] a turbulent shear boundary layer,[10] a Blasius laminar boundary layer with dominant
balance of bulk and boundary layer dissipation[6] etc seem to be able to obtain the observed
scaling.
The Prandtl number dependence of Nu at high Pr is not clear. The only studies are that of
Goldstein[5] using naphthalene mass transfer technique at Schmidt number of 2750 and of Askenazi
and Steinberg[1] at Pr of 93. The Ra1/3 scaling was observed by Goldstein till Ra < 1013 , while
Askenazi’s studies seem to support an exponent less than 1/3. At very high Pr, as the viscous
boundary layers reach their limiting thickness, it is expected that Nu becomes independent of Pr. [7]
In this case the flux is expected to follow the 4/3 law as the strength of large scale flow reduces with
increasing Prandtl Number. Hence, in adition to the large number of unresolved issues about the
1
Figure 2:
SEM
man TM membrane
image
Cdc/2
of
Pall
Gell-
lm
M
Figure 1: Experimental setup
Figure 3: Diffussion drop across the membrane
nature of high Rayleigh number turbulent free convection, few studies exist for the high Prandtl
Number regime. Further, very little is known about the nature of near wall coherent structures in
this regime.
We study high Ra turbulent free convection driven by density difference across a thin permeable horizontal partition separating two tanks of square plan form cross sections. The gravitational
potential due to a heavier fluid(brine) above a lighter fluid(water) across the partition drives the
flow, which is resisted by the presence of the micro porous partition. At low pore sizes in the
membrane, the transport across the partition would become diffusion dominated, while the transport above and below the partition becomes similar to turbulent convection above flat horizontal
surfaces. As molecular diffusivity (D) of NaCl is about 100 times lower than temperature diffusivity, larger values of Ra and Sc are achieved through this arrangement for similar driving density
potentials. The structure of convection in this case can easily be visualised. In this paper we
report the flux scaling and the nature of near wall coherent structures in high Rayleigh number
(∼ 109 to 1013 ) high Schmidt number( ∼ 650) turbulent convection.
2.
EXPERIMENTAL SETUP AND MEASUREMENTS
A schematic representation of the experimental setup is shown in Figure 1. The test section,
made of 8mm float glass, has two vertical compartments,with ground mating edges and a permeable
partition fixed in between them. Figure 2 shows a (1000×) scanning electrode microscope (SEM)
image of Pall Gelmann TM NX29325 membrane disc filters used in the experiments. These are
bilayer membranes made of nylon66 with 0.45µ mean pore size and thickness of 142.24µ. The
solidity of the membrane was calculated as 0.6 from the SEM images by computing the occupied
fraction of pixels of a binary image generated from figure 2, based on a suitably chosen threshold.
The top tank is filled with brine after the bottom tank is filled distilled water. To reduce
initial mixing, a temporary tank with sponge bottom is kept over the membrane while the top
tank is filled, the removal of which initiates the experiment. A thin transparent plexiglass sheet
floating over the brine level prevents evaporation and produces similar boundary conditions for the
two compartments. The side glass compartments hold distilled water to reduce excessive refraction
2
of the laser beam during visualisation. The assembly is mounted on a levelling table so that the
partition can be made horizontal. Test section cross sections of 15 cms× 15cms, and 10cms ×
10cms with height 23 cms, ie aspect ratios of 0.652 and 0.435 were used.
The flux is estimated from the transient measurement of the the top tank concentration.
Assuming that the fluid in both the compartments is well mixed in the region away from the
0
partition, and using the mass balance at any time CT (t) VT + CB (t) VB = CT0 VT + CB
VB , with
0
CB = 0, the concentration difference is
∆C(t) = CT (t) − CB (t) = (1 +
VT
VT
)CT (t) − CT0
VB
VB
(1)
Here, VT is the top tank solution volume, VB the bottom tank solution volume, CT (t) is the top
tank concentration, CB (t) is the bottom tank concentration and the superscript 0 denotes initial
values. The nett flux of NaCl across the partition at any instant is given by the rate of change of
top tank concentration as
d CT
q = −h
(2)
dt
where h is the top compartment height. Hence, ∆C and flux can be calculated from the transient
measurement of concentration of top tank fluid.
The concentration of NaCl in the top tank is estimated from the measurement of electrolytic
conductivity of the top tank fluid. The conductivity measurements are made by ORION SENSORLINK TM PCM100 conductivity measurement system,[9] with a 2 electrode conductivity cell, model
ORION TM 011050, with automatic temperature compensation. The probe was calibrated before
each experiment. As calculation of dc
dt directly from the measured CT vs t distribution results in
T
excessive errors due to data noise an exponential decay fit is used calculate the derivative dC
dt .
The plume structure was visualised by Laser Induced fluorescence of Sodium Flourescein dye.
The bottom tank solution was tagged with the dye and a horizontal laser sheet(Spectra-Physics,
Stabilite 2017TMAr-Ion,5W ) was passed just above the partition. The dye in the bottom solution,
while convecting upwards, fluoresces on incidence of the laser beam to make the plume structure
visible. The quantity of dye (1.2ppm) was chosen to give sufficient fluorescence intensity without
affecting the measured conductivity. The fluorescence images were captured on a digital camera
(SONY PCR 9E) after cutting off the laser light wave lengths using a yellow glass filter, Coherent
optics OG-515.
3.
3.1
RESULTS AND DISCUSSION
Flux
To compare the scaling of flux with that of high Rayleigh number turbulent free convection,
the driving potential in the present case is corrected for the concentration drop occuring across the
partition, as shown in figure 3. This concentration drop is ∆cm = q lm /(Γ D) where Γ = open area
factor of the membrane, lm = the thickness of the membrane. Therefore, effective concentration
difference on one side of the partition ∆cdc /2 = (∆c − ∆cm ) /2. We look at the scaling of flux
with (∆ρ/ρ)dc /2 = β∆Cdc /2 in the form of an alternate representation of flux. Hereafter, the
subscript dc denotes this corrected effective concentration difference.
Figure 4 shows the variation of flux with the effective driving potential obtained in the experiments. The calculations were conducted after correcting the measured concentration curve
for systematic shifts due to change in probe linearity. The error bars, which include random and
systematic errors, for flux and ∆C shown in the plot indicate the extent of possible variation.
The plot includes the results for experiments with a starting concentration of CT0 = 10g/l at
the two aspect ratios of 0.652 and 0.435, as well as experiments with starting concentration of
3
−1
0.5
Goldstein
Xia
AR=0.65
AR=0.43
Goldstein
Xia
AR=0.65
AR=0.43
0.45
0.4
10
flux(mgs/cm2/min)
0.35
0.25
δ
Ra−1/3
0.3
0.2
0.15
−2
10
0.1
0.05
1/2∆ρ/ρdc
0
0
−3
10
0.5
1
1.5
2
Ra
dc
Figure 4: Variation of flux with
1
2
∆ρ
ρ dc
2.5
11
x 10
Figure 5: Variation of Raδ−1/3 withRadc
CT0 = 3g/l for AR=0.652. Therefore, within the present accuracy of the measurements, the flux
seems to be independent of the starting concentration and the aspect ratio. The figure also shows
the corresponding results of Goldstein[5] and Xia.[8]
Theerthan[12] have shown that a more appropriate representation of non dimensional flux in
−1/3
turbulent free convection is in terms of Raδ
, where Raδ is the Rayleigh Number based on the
diffusion layer thickness, δd . The diffusion layer thickness for the current case is calculated as
−1/3
δdc = (D∆cdc /2) /q. Raδ
can be written as a ratio of two fluxes as,
−1/3
Raδ
= q/
D∆Cdc /2
Zw
,
where, Zw =
νD
gβ∆Cdc /2
1/3
(3)
is a near wall length scale for turbulent free convetion.[11] This representation does not have the
layer height as a parameter and hence is a better representation in turbulent free convection where
−1/3
near wall phenomena decides the flux. Raδ
varies only between 0.1 and 0.3 for a wide range of
Ra and various types of free convection, being a reflection of the fact that heat flux in turbulent
free convection for a given ∆T and fluid does not vary much. It could be noticed from (3) that
−1/3
Raδ
is a constant if flux scales as ∆C 4/3 .
−1/3
Figure 5 shows the variation of Raδ
with Rayleigh number using the effective concen−1/3
tration difference, Radc for the same experiments as in figure 4. The line Raδ
= 0.166 is
obtained from the correlation N u = 0.066Ra1/3 of Goldstein[5] at Pr=2750. The relation of Xia
−1/3
, Nu=0.14Ra0.297 P r−0.03 is also plotted in the figure. The current experimental values of Raδ
are nearly constant implying that the flux scales as the 4/3rd power of ∆Cdc . Goldsteins value of
−1/3
Raδ
falls within the error range of the current experiments. The deviation from the constant
−1/3
Raδ
for very low Radc seen in figure 5 cannot be inferred to be from the change in the flux
scaling, as the errors involved in calculating ∆ρ
ρ becomes large when the concentration differences
between the tanks tend to zero. Therefore, we conclude from the present experiments that Rayleigh
Numbers of 1011 − 1010 at Schmidt number of 650, the flux scales very nearly as ∆C 4/3 , and the
effect of Schmidt number on flux is negligible. To get a better understanding of the phenomena,
we now study the near wall dynamics of coherent structures in the current experiments.
4
Figure 7:
Single large scale flow cell at
AR=0.435 (a)Horizontal planform of plume
structure at Radc = 4.068 × 1011 , ∆ρ
=
ρ dc
4.515 × 10−3 Image size is 10cms × 10 cms
(b)Vertical plume structure at Radc = 4.1003 ×
= 4.551×10−3 Image size is 7.25cms
1011 , ∆ρ
ρ dc
× 4.67 cms (c)Velocity distribution for(b)
Figure 6: Multiple large scale flow cells
at AR=0.65 (a)Horizontal planform of plume
structure at Radc = 4.07 × 1011 , ∆ρ
=
ρ dc
4.52 × 10−3 . Image size is 15cms × 15 cms
(b)Vertical plume structure at Radc = 4.06 ×
1011 , ∆ρ
= 4.5 × 10−3 image size is 15cm ×
ρ dc
6.7 cms (c)Velocity distribution for(b)
3.2
Coherent structures
Figure 6(a) shows the plan form plume structure, obtained when the upcoming plumes intersect a horizontal laser sheet very near(<1mm ) the membrane. The image shows the full test section
−3
cross setion of 15cm × 15cm (AR = 0.65) and is at Radc = 4.0711 × 1011 and ∆ρ
ρ dc = 4.6 × 10
, where the flux follows 4/3rd law (see figure 5). The figure shows that the near wall coherent
structures in high Rayleigh number convection are line plumes. We notice that the plume structure display circular patches of aligned lines originating from a plume free area in the center. The
5
aligned nature of the plume lines implies the presence of a near wall flow along these lines. The
predominant motion of the fluid in these patches, as observed from the video images, were outward
from the plume free circular area along the plume lines. The lateral shift of the plumes lines,due to
the entrainment velocity of the nearby line plumes, was also very low (∼1mm/s). Hence we infer
these cells as the signatures of large scale flow cells. The large scale flow impinges on the plume
free circular area in the center of each aligned plume region and create an outward near membrane
flow which aligns the line plumes. The current image shows two large scale flow cells on the right
side of the image. These large scale flow cells shift their position randomly, implying that the large
scale flow direction in high Ra convection, at least for higher AR, is not constant.
The above picture would be more clear from Figure 6(b) which shows the plume structure in a
vertical plane at 2cms from a side wall of the test section,( ie 2cms from the bottom of figure 6(a)).
Note that Figure 6(a) is after 11.5 minutes from figure 6(b), so that the latter figure is not the exact
vertical view of the former. The height of figure 6(b) is 6.7 cms. The figure shows that plumes
combine giving rise to columns of upward rising fluid, which results in a downward travelling
portion of fluid in between the columns, which impinges on the wall and create the near wall
shear. Figure 6(c) shows the near wall velocity field vectors(overlaid over the velocity magnitude)
estimated using the spatial intensity correlation technique[4] between figure 6(b) and another image
0.4 seconds later. The upward moving colums and the resulting downward impingement in between
them could be noticed from the figure. The column rise velocity is about 0.3 cms/s while the
downward velocites are much larger (∼ 0.8 cms/s). It is known that the large scale velocity
1/3
scales as the Deardorff’s velocity scale W∗ = (gβqh) . [3] W∗ for Figure 6(c)is 0.31 cms/s, of the
same order as the plume rise velocity. Hence, one could infer that the large scale flow velocity is
essentially driven by plume columns, which inturn organise the plume structure. The current study
shows that the mean shear near the walls could be larger than that due to W∗ as the downward
velocities between the plume columns are higher than W∗ .
The planform plume structure at AR=0.435 is shown in figure 7(a). The image size is the
same as the test section cross section of 10cms × 10 cms. Figure 7(b) shows the vertical plume
structure 13 min prior to figure 7(a). The image shows a vertical section(normal to plume structure
in figure 7(a)) 2cms above the bottom side of figure 7(a) of 7.25cms width and 4.67 cms height
starting from the left of figure 7(a). The plan form plume structure and the vertical image lead
us to the inference that there is a single large scale flow cell spanning the full tank cross section.
The large scale flow impinges on the membrane on the right and creates a near membrane velocity
from right to left. This results in inclined plumes near the membrane as seen in the vertical image.
Plumes combine and rise along the left wall, feeding the mean circulation, which sustains the mean
shear near the membrane. The velocity estimates in figure 7(c) from spatial correlation between
images 0.4 seconds apart reproduces the dynamics reasonably well, with the plume column rise
velocity at the LHS of the figure(∼ 0.3cms/s) being of the same order as the large scale flow velocity
estimated as the Deardorff scale (0.31cms/s).
4.
CONCLUSIONS
Experiments show that the flux scaling in high Rayleigh Number -high Schmidt number
turbulent convection, even in the presence of a large scale flow, follows the 4/3 rdlaw and seems
to have little dependence on Sc. The flux is only weakly dependent on the AR. The near wall
coherent structures are line plumes. We detect multiple large scale flow cells and changing large
scale flow direction at higher AR. Lower AR shows a single large scale flow cell and constant sense
of near membrane mean shear. The large scale flow is shown to be sustained by rising columns of
combined plumes, the velocity of which scales as Deardorff scale.
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